# Sharpness of the phase transition in percolation models

- 333 Downloads
- 90 Citations

## Abstract

The equality of two critical points — the percolation threshold*p*_{ H } and the point*p*_{ T } where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneous*d*-dimensional lattices (*d*≧1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter*M*(β,*h*), which for*h*=0 reduces to the percolation density*P*_{∞} — at the bond density*p*=1−*e*^{−β} in the single parameter case. These are: (1)*M*≦*h*∂*M*/∂*h*+*M*^{2}+β*M*∂*M*/∂β, and (2) ∂*M*/∂β≦|*J*|*M*∂*M*/∂*h*. Inequality (1) is intriguing in that its derivation provides yet another hint of a “ϕ^{3} structure” in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents\(\hat \beta\) and δ. One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound δ≧2, and the other implies the result of Chayes and Chayes that\(\hat \beta \leqq 1\). An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation\(\hat \beta (\delta - 1) \geqq 1\) and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.

## Keywords

Neural Network Phase Transition Long Range Cluster Size Quantum Computing## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys.
**74**, 41–59 (1980)Google Scholar - 2.Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb.
**56**, 229–237 (1981)Google Scholar - 3.Aizenman, M., Newman, C.M.: Discontinuity of the percolation density in one-dimensional 1/|
*x−y*|^{2}percolation models. Commun. Math. Phys.**107**, 611–647 (1986)Google Scholar - 4.Aizenman, M., Chayes, J.T., Chayes, L., Imbrie, J., Newman, C.M.: An intermediate phase with slow decay of correlations in one-dimensional 1/|
*x−y*|^{2}Ising and Potts models (in preparation)Google Scholar - 5.Aizenman, M.: Contribution in: Statistical physics and dynamical systems (Proceedings Kösheg 1984). Fritz, J., Jaffe, A., Szasz, D. (eds.). Boston: Birkhäuser 1985Google Scholar
- 6.Chayes, J.T., Chayes, L.: Critical points and intermediate phases on wedges of ℤ
^{d}. J. Phys. A (to appear)Google Scholar - 7.Hammersley, J.M.: Percolation processes. Lower bounds for the critical probability. Ann. Math. Statist.
**28**, 790–795 (1957)Google Scholar - 8.Aizenman, M., Newman, C.M.: Tree graph inequalities and critical behavior in percolation models. J. Stat. Phys.
**36**, 107–143 (1984)Google Scholar - 9.Newman, C.M., Schulman, L.S.: One-dimensional 1/|
*j − i*|^{s}percolation models: The existence of a transition for*s*≦2. Commun. Math. Phys.**104**, 547–571 (1986)Google Scholar - 10.Griffiths, R.B., Hurst, C.A., Sherman, S.: Concavity of magnetization of an Ising ferromagnet in a positive external field. J. Math. Phys.
**11**, 790–795 (1970)Google Scholar - 11.Newman, C.M.: Shock waves and mean field bounds. Concavity and analyticity of the magnetization at low temperature. Appendix to contribution in Proceedings of the SIAM workshop on multiphase flow, G. Papanicolau (ed.) (to appear)Google Scholar
- 12.Aizenman, M., Fernández, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys.
**44**, 393–454 (1986)Google Scholar - 13.Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.: Renormalization group approach to percolation problems. Phys. Rev. Lett.
**35**, 327–330 (1975)Google Scholar - 14.Chayes, J.T., Chayes, L.: An inequality for the infinite cluster density in Bernoulli percolation. Phys. Rev. Lett.
**56**, 1619–1622 (1986)Google Scholar - 15.Fröhlich, J., Sokal, A.D.: The random walk representation of classical spin systems and correlation inequalities. III. Nonzero magnetic field (in preparation)Google Scholar
- 16.Fernández, R., Fröhlich, J., Sokal, A.D.: Random-walk models and random-walk representations of classical lattice spin systems (in preparation)Google Scholar
- 17.Griffiths, R.B.: Correlations in Ising ferromagnets. II. External magnetic fields. J. Math. Phys.
**8**, 484–489 (1967)Google Scholar - 18.Aizenman, M., Kesten, H., Newman, C.M.: Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Submitted to Commun. Math. Phys.Google Scholar
- 19.van den Berg, J., Kesten, H.: Inequalities with applications to percolation and reliability. J. Appl. Probab.
**22**, 556–569 (1985)Google Scholar - 20.Kesten, H.: Percolation theory for mathematicians. Boston: Birkhäuser 1982Google Scholar
- 21.Griffeath, D.: The basic contact process. Stochastic Processes Appl.
**11**, 151–185 (1981)Google Scholar - 22.Aizenman, M., Barsky, D.J., Fernández, R.: The phase transition in a general class of Ising-type models in sharp. Submitted to J. Stat. Phys.Google Scholar
- 23.Chayes, J.T., Chayes, L., Newman, C.M.: Bernoulli percolation above threshold: an invasion percolation analysis. Ann. Probab. (to appear)Google Scholar